For how to root android phones without computer, there are plenty of Android apps available in the google play store such as FramaRoot, vRoot etc. you can try any of these Androids apps and root your device without using a computer. However, before going forward remember to take a complete back up of your Android phone, ensure that the battery percentage on your android phone is more than 50 and read the instructions on how to root android using the app and begin the rooting process.
The Pros to root an android device are
-you can access all files Android Files and Folders,
-Increase the performance of the Android phone by overclocking,
-Increase the battery life of the android phone,
-ability to Install custom ROMs,
-get access to Root Apps and
-Increase the RAM memory.
The Cons to root a android device are
-your phone might just be a hard brick if rooting fails,
-all data loss,
-phone warranty void,
-you will get random phone restart and
-the security/privacy is no more